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Bootstrapping frustrated magnets: the fate of the chiral ${\rm O}(N)\times {\rm O}(2)$ universality class
by Marten Reehorst, Slava Rychkov, Benoit Sirois, Balt C. van Rees
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Submission summary
Authors (as registered SciPost users):  Marten Reehorst · Slava Rychkov 
Submission information  

Preprint Link:  https://arxiv.org/abs/2405.19411v2 (pdf) 
Date submitted:  20240704 14:28 
Submitted by:  Reehorst, Marten 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study multiscalar theories with $\text{O}(N) \times \text{O}(2)$ symmetry. These models have a stable fixed point in $d$ dimensions if $N$ is greater than some critical value $N_c(d)$. Previous estimates of this critical value from perturbative and nonperturbative renormalization group methods have produced mutually incompatible results. We use numerical conformal bootstrap methods to constrain $N_c(d)$ for $3 \leq d < 4$. Our results show that $N_c> 3.78$ for $d = 3$. This favors the scenario that the physically relevant models with $N = 2,3$ in $d=3$ do not have a stable fixed point, indicating a firstorder transition. Our result exemplifies how conformal windows can be rigorously constrained with modern numerical bootstrap algorithms.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
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 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
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Report
The authors study phase transitions in the frustrated magnets, which
are described by a matrix model with symmetry O(N)XO(2). They use the
conformal bootstrap method and find that no unitary theory
can be found in $d=3$ for N smaller than some value that they evaluate
at 3.78.
The results obtained in this article are interesting and deserve
publication in scipost. There is however few points I would like to
raise.
1) In sect. 3.3.2 the authors study the fate of the fixed points C+
and C, which they relate to minima of their navigator function, as N
is decreased. They find that C+ "turns into a saddle". As far as I
understand, the navigator function is continuous and there is a finite
island where it is negative. Consequently, if C+ becomes a saddle
point, local minima must appear at the same time (in agreement with
Morse theory). What happens to these minima?
2) If the minimum C+ disapears around N=7, I would expect that, for
$3.76<N<7, the island found by the authors corresponds to an unstable
fixed point in the renormalizationgroup sense. The phase
transition would be generically of first order except at some
tricritical point and Nc(3) would be close to 7, not
3.76. Unfortunately, the authors are unable to compute the scaling
dimension of the operator SS' and cannot really test this. An
alternative would be that it is not always true that a local minimum
can be associated with a fixed point. I would encourage the authors to
comment on this.
3) In Fig 8, the curves representing the scaling dimensions of various
operators clearly show a feature around 4.2 (a breaking of the slope
or maybe a square root). What is at the origin of this behavior?
4) Finally, I think that the references 45 and 46 should appear earlier
in the introduction.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
This paper considers the merger and annihilation of CFTs with O(N)xO(2) symmetry in d=3, which describe certain statistical systems. Using the conformal bootstrap, they find an island of an allowed region that matches perturbative results at large N and near d=4, and they compute how this island disappears as a function of d and N. In particular, they find that it disappears for d=3 for N>3, which excludes the most physically interesting cases of N=2,3. The island disappears likely due to violation of unitarity as the CFT goes into the complex plane, this violation of unitarity is much more significant than the violation of unitarity from being in fractional N and d, which as they discuss is not significant enough to affect bootstrap results as shown in previous work. Near the point where the island disappears, they also find that CFT data shows a square root behavior as expected from merger and annihilation.
The paper is excellently written and all calculations seem correct, so I suggest no changes. Looking ahead, it would be nice to also bootstrap the model with two relevant singlets (this paper only focussed on the model with one relevant singlet), so that they can explicitly see the merger and annihilation of the two fixed points.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)